Reliability analysis of tunnel convergence: Is it worth

the effort? Faustino, G.*1, Bilé Serra, J. *2

1 Laboratório Nacional de Engenharia Civil, Lisbon, Portugal

(presently at COBA Consulting Engineers) 2 Laboratório Nacional de Engenharia Civil, Lisbon, Portugal

ABSTRACT The safety of tunnel excavation is generally evaluated by means of deterministic approaches. The complexity of the pheno-mena mobilized by tunnel excavation has favored the adoption of simplified methods of analysis within which the convergence-

confinement method is frequently preferred. This analytical method was used to perform a reliability analysis of the influence of two key

design parameters, namely the distance lag at the support installation position and the stability limit of convergence through the analysis of

a case study. Reliability analysis was performed using FORM methodologies in order to approximate the limit state functions. These func-

tions represent the two main radial failure mechanisms, i.e. the resistance of the support, when the maximum pressure allowed is mobilized,

and the excessive convergence of the cavity, which is assessed with limit values for deformation. The results are considered separately and together. The possibility of the applicability of finite element methods within this methodology is also investigated, using the software

PLAXIS.

1 INTRODUCTION

The safety of tunnel excavation is typically evaluated

by means of deterministic analyses. The complexity

of the phenomena mobilized by tunnel excavation

and the paucity of geological and geotechnical data

has favored the adoption of simplified methods of

analysis in the past.

From the geotechnical point of view conventional

concepts such as the global safety factor are still used

for the purpose of assessing the safety level. It is

based on the experience of generations of both design

and site engineers, having many times proved suc-

cessful and led to adequate safety standards. Howev-

er, a formal weakness of this concept is that the un-

certainty is not associated with the real causes. Since

the uncertainty and variability of the ground and the

randomness of support properties play a significant

role in the reliability level achieved with a given de-

sign, the issue of the applicability of probabilistic

based methods in tunnel design is gaining relevance.

There have been several probabilistic analysis of

underground excavation of stability problems (e.g. Li

and Low, 2010 or Oreste, 2005). On the subject of

tunnel support, Schweiger et al. (2001) carried out a

probabilistic reliability analysis with a finite element

code. Albeit less frequent, reliability evaluation of

serviceability performance, e.g. subsidence, has also

been addressed in the past (Mollon et al., 2009 and

Serra and Miranda, 2013).

This paper addresses the reliability evaluation of

different modes of failure of a deep tunnel by study

the influence of both distance from the tunnel face of

the support position and different limit failure values

of the probability of failure. These calculations were

performed using a simplified methodology to analyze

the interaction between the ground and the support -

the Convergence-Confinement Method – commonly

1

used in preliminary design phases. It was also inves-

tigated with a brief example if more representative

analysis methods such as finite element methods

(FEM) can be used to perform reliability analysis.

2 CONVERGENCE-CONFINEMENT METHOD

2.1 General considerations

The convergence-confinement (CC) method, i.e.

the characteristic curve method, was developed to

simulate three dimensional effects caused by the ex-

cavation of a tunnel by means of a simplified two

dimensional analysis. Strictly speaking, the validity

of this method is restricted to deep tunnels where the

prevailing failure mechanism is related to the con-

vergence of the cavity (Panet and Guellec, 1974). In

an ingenious but simple way, these effects can be ac-

counted for by considering a fictitious inner pressure

at the tunnel wall, that in a plane deformation analy-

sis would correspond to a given convergence value

(Figure 1).

Figure 1. Fictitious inner pressure on the support and confinement

loss ratio (Martin, 2012)

With this information it is possible to construct the

longitudinal displacement profile (LDP) which aims

at including the three dimensional effects in a simpli-

fied two dimensional analysis. It consists on a dia-

gram that represents the radial displacement of the

tunnel wall along the axis. In order to construct a rig-

orous analytical profile it is necessary to carry out a

three-dimensional analysis. However, several analyt-

ical solutions to construct this curve are available.

The one by Vlachopoulos and Diederich (2009) was

used in this paper.

The convergence-confinement method requires the

construction of three basic curves, addressed in detail

in several references, such as in Lü et al. (2011). The

first one is the longitudinal deformation profile which

was already described. The other two will be ad-

dressed bellow.

2.2 Characteristic curves of the ground and of the

liner

The ground reaction curve (GRC), or characteris-

tic curve, defines the relationship between the inner

pressure decrease from the geostatic condition and

the radial displacement at the tunnel wall. As the

tunnel face advances and moves away from the anal-

ysis section, the fictitious support pressure decreases.

As a consequence, the displacement will gradually

increase, as does the plastic radius, whereas the sup-

port pressure at equilibrium tends to reduce.

The vast majority of the closed form solutions as-

sume elastic-plastic mechanical response with either

Mohr-Coulomb or Hoek-Brown failure criterion. In

this paper the Duncan Fama (1993) analytical solu-

tion, which is based on the Mohr-Coulomb failure

criterion, was used to construct the GRC.

The support characteristic curve (SCC) defines

the dependence between the pressure at the support

system and its inward radial displacement. If one

admits an elastic/perfectly-plastic response of the

support, it takes only two parameters to fully describe

it, i.e. (i) the elastic stiffness ks, which is the ratio be-

tween the radial pressure increment and the variation

of radial deformation, and (ii) the resistance of the

support Psmax, beyond which the support yields and

fails. A common design criterion is to limit the max-

imum support load to a fraction of the resistance, that

is, to define a limit value to the deformation.

The elastic stiffness and the limit pressure were

determined following Hoek et al (2002).

2.3 Attaining equilibrium

As illustrated in Figure 2, one can simulate the in-

teraction between the support and the ground and de-

termine the equilibrium pressure (Peq) and displace-

ment (ueq).

Prior to the support installation, a displacement uin

(which can be determined by the LDP) has already

occurred due to the confinement loss. The support is

then installed and starts to deform until it attains both

2

equilibrium pressure and displacement compatibility

with the ground at point A.

Figure 2. Ground-support interaction (Oreste, 2002)

3 RELIABILITY ANALYSIS OF TUNNEL

EXCAVATION

3.1 Level II reliability analysis

Level II methods are based on linear (1st order) or

quadratic (2nd order) approximations of the limit state

function (LSF). The former are named as FORM

(First-Order Reliability Methods) (see Figure 3).

The probability of failure is quantified by means

of a reliability index β, which corresponds to the dis-

tance between the center of the normalized space and

the design point, which is the most probable point in

case of failure. This is the point of first tangency in

the space of the representative standard variables

(e.g. U1 and U2 in Figure 3) between the origin cen-

tered circles and the limit state function.

Figure 3. FORM procedure in standardized space (Nadim, 2007)

3.2 Limit state functions

Two performance functions were considered, con-

cerning the structural resistance of the support

(g1(X)) and the maximum convergence allowed in

the tunnel walls of radius R (g2(X)).

g1(X)=

Psmax

Peq

-1 [1]

g2(X)= εmax-

μeq

R [2]

where εmax is the limit of the radial linear strain and

ueq stands for the inward radial displacement at equi-

librium and the subscript eq stand for “in equilibri-

um”.

3.3 Computational implementation of the

reliability analysis

The FORM methodology was implemented using

an alternative interpretation of the Hasofer-Lind reli-

ability index (βHL), proposed by Low and Tang

(2001). The methodology is described in detail in

their work.

A continuous model of the ground by means of an

elastic/perfectly-plastic deformation model obeying

the Mohr-Coulomb failure criterion with isochoric

deformation was assumed. The corresponding three

random parameters are the deformability modulus

(E’), the friction angle ϕ’ and the apparent cohesion

c’, expressed in terms of effective stress components

all assumed to follow a Gaussian probability distribu-

tion with parameters μi and σi and correlation matrix

R. Regarding the support, the only discrete parameter

is its resistance to radial pressure action.

The design point xd, i.e. that on the limit state

function with the smallest distance to the origin, is

obtained by means of the minimization of the relia-

bility index (β) as defined by Ditlevensen (1981).

β=min√[xi-μi

σi]

T

[R]-1

[xi-μi

σi] [3]

The MS Excel solver add-in is used in a minimiza-

tion problem of β considering a few additional con-

straints, namely:

1) g(X)=0, concerning the fact that the solution

vector (design point) has to verify the condi-

tion of the limit state function

2) 𝑃𝑒𝑞 ≤ 𝑃𝑠𝑚𝑎𝑥; and

3) 𝑢𝑔𝑟𝑜𝑢𝑛𝑑 = 𝑢𝑠𝑢𝑝𝑝𝑜𝑟𝑡

This process is addressed in detail in Faustino

(2013).

3

4 A DEEP TUNNEL IN SOFT ROCK

4.1 Ground Properties

The example used to illustrate the reliability anal-

ysis of ground-support interaction (using the conver-

gence-confinement method) consists of a circular

tunnel of radius equal to 5 m under a hydrostatic

stress field of 7 MPa.

The properties are listed in Table 1.

Table 1. Statistical parameters for the variables

Parameters μi 𝜎i CoV

(%)

E′ [MPa] 1800 500 28

ϕ′ [deg] 23 3.3 22

c’ [kPa] 1500 320 14

The Poisson’s ratio of the rock mass is assumed as

deterministic and equal to 0.3.

4.2 Support properties

The support is also regarded as deterministic. A

shotcrete lining 25 cm thick is considered. The

adopted mechanical properties are as follows:

Young’s modulus E=30 GPa with an uniaxial com-

pression strength of 35 MPa and a Poisson’s ration of

υ=0.2 . The stiffness and support capacity are estimat-

ed using the equations proposed by Hoek et al

(2002). The input parameters result in psmax =1706

kPa and ks=320.7 MPa/m.

4.3 Reliability Analysis

The reliability analysis was performed analyzing

each limit state function individually. In order to ac-

count for the effects of both limit state functions and

to create a more realistic global failure criterion, in

terms of probability of failure, the results of both

curves were jointly considered by retaining the great-

er probability value of each individual function for

each scenario analyzed.

The probability of failure versus lag distance of

support installation (up to 15 m behind the face) was

studied. The effect of the convergence limit on the

probability of failure’s surface (LSF g2(X)) and con-

sequently in the global results was also addressed.

The limit value for the radial linear strain conver-

gence was set in 0.5%, 0.75%, and 1%. A value of

0.2% was also analyzed but no results were found

since the support properties were not compatible with

the equilibrium attainment with such a strict strain

requirement. The results are shown below.

Limit state function 1 (g1(X)):

The probability of failure of the support decreases,

as expected, with the unsupported distance and

becomes negiglable for distances above 2 m, i.e. D/5

(Figure 4). This is a probability based illustration of

the NATM principle that favours ground deformation

in order to reduce the support load.

Figure 4. Probability of failure and reliability index Vs distance

from the tunnel face of the support position

Limit state function 2 (g2(X)):

Again accordingly with the NATM principles, the

probability of excessive convergence evolves in the

opposite sense from the probability of support fail-

ure. The importance of and the sensitivity to the

strain limit criterion (Eq. 2) is illustrated in Figure 5.

Figure 5 & 6. Probability of failure and reliability index vs dis-

tance from the tunnel face of the support position (εmax=1% ; εmax=

0.75%; εmax= 0.5%).

One can appreciate the reliability achieved with a

given limit value, one of the three close values of

0

0,2

0,4

0,6

0,8

1

0 1 2 3 4 5P

rob

ab

ilit

y o

f fa

ilu

reDistance from the tunnel face (m)

0

0,2

0,4

0,6

0,8

1

0 5 10 15

Prob

ab

ilit

y o

f fa

ilu

re

Distance from the tunnel face (m)

εmax=1%εmax=0.75%εmax=0.5%

-4

-2

0

2

4

0 5 10 15

Rel

iab

ilit

y i

nd

ex β

Distance from the tunnel face (m)

4

0.5%, 0.75% and 1%, considering different unsup-

ported length values. As an example, let one admit

that a minimum reliability index of 3 is required for

safety reasons. For the assumed geotechnical condi-

tions, one can observe that one should restrict the

limit strain value to either 0.75% or 1%. Otherwise,

the limit strain criteria (0.5%) would be violated too

frequently, causing the implementation of corrective

measures, e.g. shortening the unsupported length.

As for the joint consideration of both limit state

function, the results below clearly suggest that the

optimum design decision for a given pair (εmax, psmax)

is attained at an intermediate distance lag. Although

this optimum distance is dependent on the pair cho-

sen, the lag distance is not quite sensitive to this

choice (Figure 8). It would seem that, with the pre-

sent scenario, it would be unsafe to construct the tun-

nel as the maximum value of the reliability index is

hardly above 2.

Figure 7. Probability of failure and reliability index vs distance from the tunnel face of the support position – g1(X) and g2(X)

(εmax= 1% ; εmax= 0.75%; εmax= 0.5%).

Yet, a remark is in order about the meaning of the

probability and reliability index values presented.

Those are not to be understood as absolute values, ra-

ther as a sorting index for the situations in presence.

One must remember that the present results corre-

spond to a given choice of mechanical model (CCM),

of constitutive models (M-C model), of statistical

distributions (Gaussian) and a solution method

(FORM). Also in a simplified way some aspects were

considered deterministic rather than uncertainties,

with proper distributions and statistical parameters.

5 APPLICABILITY TO FEM

In order to test the applicability of this methodology

to finite element methods, a simple example was

considered, regarding the same problem but consider-

ing only the LSF 1, evaluated for a lag distance of

𝐿 = 2𝑚. The results of the analytical evaluation us-

ing CCM (see Figure 4) are listed in Table 2.

The response surface method (RSM) (Pula and

Bauer, 2007) was used to build the LSF and the re-

sponses needed to calibrate this function were ob-

tained using the commercial software PLAXIS v8.

Table 2. Results of the analytical analysis

Function c’ (kPa)

ϕ’ (°)

E

(MPa)

β pf

LSF1 1950.99 22.2 487.7 2.98 0.00146

Figure 8. Finite Element Method Mesh

The tunnel boundary was simulated using 60 line

segments and the ‘Total Multiplier’ modulus was

used to apply the required pressure on the tunnel

walls. The finite element mesh is represented in Fig-

ure 8.

A total of 3 iterations with 27 calculations each were

performed. The Eq. 4 shows the resulting limit state

function and the Table 3 the design point and the cor-

responding reliability index (β) and probability of

failure.

g1(X)=-0,129-5,4×x10-4∙c'+1,6×10-7

∙ c'2+4,7579∙ tan ϕ' -4,599∙ tan ϕ'2

-4,1×10-4∙E+5,1×10-8∙E2

-9,5×10-4∙c'∙ tan ϕ'+ 1,8×10-7∙c'∙E

+7,4×10-4∙E∙ tan ϕ'

[4]

Table 3. Results of the numerical analysis (using RSM)

Function c’ (kPa)

ϕ’ (°)

E

(MPa)

β pf

LSF1 1877.62 22.1 555.5 2.74 0.003

Comparing the results from analytical and numeri-

cal analysis it can be concluded that both results are

fairly close to each other not only regarding reliabil-

ity index and probability of failure but also in the

5

way the design point vector deviates relatively to the

midpoint.

The detailed explanation and the full set of results

can be found in Faustino (2013).

6 CONCLUDING REMARKS

The effects of the distance from the support instal-

lation have been investigated regarding reliability

analysis for two main issues, each one resulting in

different limit state function.

The results show that different consequences re-

sult from the range of distances of the support instal-

lation and that the choice of which results in a lower

probability of failure isn’t simple when analyzed

simply a function at a time.

In this example, it is clear that a better understand-

ing of the global effects can be obtained if both at-

tained surfaces are analyzed together as each one

evolves in a different way with the increasing of the

distance.

For small distances, the support shapes the global

surface as the ground pressure on the support remains

high (and equal to its capacity). With the increase of

this distance the importance of the displacements be-

comes higher, eventually shaping the global probabil-

ity of failure’s surface.

If both the support and the displacements of the

ground are addressed together it is also clear that a

small gap between the tunnel face and the support

position considerably reduces the risks of failure

(collapse).

The numerical analysis proved to be a viable al-

ternative, for this case, when used together with the

Response Surface Methodology (RSM) approach.

Given the relatively simplicity of this analysis and

the importance of the information that can be ob-

tained, we believe that the answer to “RELIABILITY

ANALYSIS OF TUNNEL CONVERGENCE: IS IT

WORTH THE EFFORT?” is undoubtedly yes.

ACKNOWLEDGEMENT

The authors would like to thank the Portuguese Ge-

otechnical Society for the sponsorship given to

present this work.

REFERENCES

Ditlevensen, O. (1981). Uncertainty modelling: with applications

to multidimensional civil engineering systems. New York,

McGraw-Hill. Dunca Fama, M.E. (1993). Numerical modelling of yield zones in

weak rocks, in: Hudson JA, editor. Comprehensive rock

engineering, pp 49-75. Oxford, Pergamon. Faustino, G. (2013). Structural safety in Geotechnics by means of

probability based methods. MSc Dissertation, Faculdade

de Ciências e Tecnologia da Universidade Nova de Lis-boa. (in Portuguese).

Hoek, E., Carranza-Torres, C. and Corkum, B. (2002). Hoek-

Brown Faiure Criterion 2002 edition. Proceedings of NARMS-TAC, pp. 267-273.

Li, H.-Z. and B.K. Low, Reliability analysis of circular tunnel un-

der hydrostatic stress field. Computers and Geotechnics, 2010. 37(1): p. 50-58.

Low, B. and Tang, W.H.(2001) Efficient reliability evaluation us-

ing spreadsheet. Proceedings of the Eight International Conference on Structural Safety and Reliability,

ICOSSAR 01, Newport Beach, California, 8 pages, A.A.

Balkema Publishers. Lü, Q., Sun, H-Y and Low, B.K. (2011). Reliability analysis of

ground-support interaction in circular tunnels using the

response surface method. International Journal of Rock Mechanics & Mining Sciences, 48 (2011), pp 1329-1343.

Martin, F. (2012). Rock Mechanics and Underground Works, 8th

edition, ENS Cachan. (in French). Mollon, G., Dias, D. and Soubra, A.H. (2009). Probabilistic anal-

yses of tunneling-induced ground movements. Journal of

Geotechnical and Geoenvironmental Engineering, 135(9), pp. 1314-1325.

Nadim, F. (2007). Tools and strategies for dealing with uncertain-

ties in geotechnics. Probabilistic Methods in Geotech-nical Engineering, Springer. pp. 71-95

Oreste, P. (2002) Analysis of structural interaction in tunnels using

the convergence– confinement approach. Tunnelling and Underground Space Technology 18. (2003), pp. 347-363.

Oreste, P. (2005) A probabilistic design approach for tunnel sup-

ports. Computers and Geotechnics, 32(7), pp. 520–34 Panet, M. and Guellec, P. (1974) – Contribution to the assessment

of the support behin the tunnel face. Proceedings 3rd In-

ternational Congress on Rock Mechanics, vol.2, part B, Denver, USA.

Pula, W. and Bauer, J. (2007). Application of the response surface

method, Probabilistic Methods in Geotechnical Engineer-ing, Springer, pp.147-168.

Schweiger, H. F., Thurner R. P. and Pöttler R. (2001) Reliability

analysis in geotechnics with deterministic finite elements. International Journal of Geomechanics, 1(4), pp.

389-413. Bilé Serra, J. and Miranda, L. (2011). The influence of ground spa-

tial variability on the settlements caused by tunnel exca-

vation. Comptes-Rendues Congrès International AFTES Espaces Souterrains de Demain, Lyon, France

Vlachopoulos N and Diederichs, M.S. (2009). Improved longitudi-

nal displacement profiles for convergence-confinement analysis of deep tunnels. Rock Mechanics and Rock Engi-

neering 42(2), pp 131-146.

6